Question

Chinovac Labs tried a new vaccine on 165 randomly selected individuals. From the experiment, it was determined that 82 of them developed immunity. Calculate a 95% confidence interval for the proportion p of individuals in the population for whom the vaccine would help. What is the lower bound of this interval?

Answer 1

$\hat{p}=\frac{82}{165}=0.49697\\\left( \hat{p}-\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}}z_{\frac{1+\gamma}{2}},\hat{p}+\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}}z_{\frac{1+\gamma}{2}} \right) =\\=\left( 0.49697-\sqrt{\frac{0.49697\left( 1-0.49697 \right)}{165}}\cdot 1.960,0.49697+\sqrt{\frac{0.49697\left( 1-0.49697 \right)}{165}}\cdot 1.960 \right) =\\=\left( 0.420678,0.573261 \right) \\Lower\,\,bound\,\,is\,\,0.420678$